My family had one that my mom sold when I was in high school. It was one that the sowing machine flipped down into the table it even survived a fire. I wish I still had it and probably would use it.
Best I can tell it's from the late 1890's. I seen one mfg from 1894 with a serial # started with a 12. This one serial number starts with 13 so best I can figure it's late 1890's some time. Anyways I think it's pretty cool and my wife loves it.
Well, I couldn't find anything easily online with that serial number, but it kinda looks like a Model 27K, which dates around 1885. You might want to send those pics in to Singer and see if they can help you.
If anybody can bring it back to life, it's you. You have more talent in the tip of your little finger than most of us have in our whole bodies. I can't wait to see the transformation.